Course:PHYS350/Tutorial 2 2007W

2. Given the following force,

$F_{x}=ax+by^{2},F_{y}=az+2bxy,F_{z}=ay+bz^{2},$

evaluate the work done in taking a particle from the origin to the point (1,1,0): (i) by moving first along the $x-$axis and then parallel to the $y-$axis, and (ii) by going in a straight line. Verify that the result in each case is equal to minus the change in the potential energy function.

4. Compute the work done in taking a particle around the circle $x^{2}+y^{2}=a^{2},z=0$ if the force is (a) $F_{x}=y,F_{y}=F_{z}=0$ and (b) $F_{x}=x,F_{y}=F_{z}=0$. Evaluate the line integrals directly and using Stokes' theorem to convert them to surface integrals.

15. A particle starts from rest and slides down a smooth curve under gravity. Find the shape of the curve that will minimize the time taken between two given points. [Take the origin as the starting point and the $z$ axis downwards. Show that the time taken is

$\int _{0}^{z_{1}}\left[{\frac {1+\left(dx/dz\right)^{2}}{2gz}}\right]^{1/2}dz$

and hence that for a minimum

$\left({\frac {dx}{dz}}\right)^{2}={\frac {c^{2}z}{1-c^{2}z}}$

where $c$ is a constant of integration. To complete the integration, use the substitution $z=c^{-2}\sin ^{2}\theta$. This famous curve is known as the brachistochrone. It is in fact an example of a cycloid, the locus of a point on the rim of a circle of radius $1/2(c^{2})$ being rolled beneath the $x-$axis.

This problem was first posed by Johann Bernoulli on New Year's Day 1697 as an open challenge. Newton's brilliant solution method initiated the calculus of variations. Bernoulli had an equally brilliant idea, using an optical analogy with refraction of a light ray through a sequnce of plantes and Fermat's principle of least time.]